Conley's fundamental theorem of dynamical systems or Conley's decomposition theorem states that every flow of a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
with
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
phase portrait
In mathematics, a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions is represented by a different point or curve.
Phase portraits are an invaluable tool in st ...
admits a decomposition into a chain-recurrent part and a gradient-like flow part. Due to the concise yet complete description of many dynamical systems, Conley's theorem is also known as the fundamental theorem of dynamical systems. Conley's fundamental theorem has been extended to systems with non-compact phase portraits and also to hybrid dynamical systems.
Complete Lyapunov functions
Conley's decomposition is characterized by a function known as complete Lyapunov function. Unlike traditional
Lyapunov function
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s ...
s that are used to assert the stability of an equilibrium point (or a fixed point) and can be defined only on the basin of attraction of the corresponding attractor, complete Lyapunov functions must be defined on the whole phase-portrait.
In the particular case of an autonomous differential equation defined on a compact set ''X'', a complete Lyapunov function ''V'' from ''X'' to R is a real-valued function on ''X'' satisfying:
* ''V'' is non-increasing along all solutions of the differential equation, and
* ''V'' is constant on the
isolated invariant sets.
Conley's theorem states that a continuous complete Lyapunov function exists for any differential equation on a compact metric space. Similar result hold for discrete-time dynamical systems.
See also
*
Conley index theory
In dynamical systems theory, Conley index theory, named after Charles Conley, analyzes topological structure of invariant sets of diffeomorphisms and of smooth flows. It is a far-reaching generalization of the Hopf index theorem that predicts ex ...
References
Theorems in dynamical systems
Differential topology
Topological dynamics
{{Math-stub